{"id":"f032c546-ad60-4cac-a5a5-807a20ed1a85","arxiv_id":"2506.09139","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"The paper defines p-adic Shimura classes and derived Hecke operators and conjectures that their action on weight-one forms equals the p-adic logarithm of a Stark unit.","lead":"This note proposes a p-adic analogue of the Harris-Venkatesh derived Hecke algebra for weight-one modular forms, introducing p-adic Shimura classes and operators on completed cohomology and conjecturing that their pairing equals a p-adic Stark regulator. The conjecture is not proved, but the paper supplies constructions and a technical theorem intended to make the conjecture precise.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"§5.1's p-adic Shimura class relies on treating ramified covers X_1(p^n)→X_0(p^n) as flat torsors; the action has nontrivial stabilizers for n>1, so no H^1_fl class is justified, leaving Conjecture 2 undefined.","rationale":"The reader's weakest assumption singled out exactly the flat cohomology step in §5.1 as the load-bearing gap, and the present stress-test agrees: the existence of S_{Z_p,fl} is not established by the argument in the paper. The concern is not a disagreement with the broader conjecture, which is honestly labeled as a conjecture; it is a correctness risk in the definition of the objects in Conjecture 2. If the covers X_1(p^n)→X_0(p^n) do not define flat torsor classes, then the p-adic norm and the p-adic derived Hecke operator do not exist, so the conjecture cannot be formulated. The proposed test directly checks torsoriality by computing stabilizers or the Čech cocycle condition at a ramified point, which would settle whether the construction is valid. The paper repeatedly defers full proofs to 'upcoming work,' so the conditional verdict is appropriate; no change to the reader's verdict is needed. The regulator Reg_{Z_p} is also only sketched, but it is secondary because the left-hand side of Conjecture 2 is already in question. The author's self-description as a research announcement is consistent with the conditional acceptance, provided the missing flat cohomology argument is supplied or the conjecture is stated with an explicit proviso.","tokens_in":10780,"tokens_out":15970,"duration_ms":162130,"concrete_test":"Fix n=2 and a prime p∤6N. Compute the geometric stabilizer of a cusp of X_0(p^2) under the natural action of (Z/p^2)^\\times on X_1(p^2), either by explicit modular curve models over C or by the mod-p description of the integral model over Z_p. If the stabilizer has order divisible by p, the map X_1(p^2) ×_{X_0(p^2)} X_1(p^2) → X_1(p^2) is not an isomorphism to X_1(p^2) × (Z/p^2)^\\times, so π_2 is not a (Z/p^2)^\\times-torsor. Then verify whether the cochain described in §5.1 satisfies the flat Čech 1-cocycle condition at the ramified point; if it does not, no element of H^1_fl(X_0(p^2), ((Z/p^2)^\\times)_a) arises from this construction, and the p-adic Shimura class is undefined.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Conjecture 2 is formulated in terms of the p-adic Shimura class S_{Z_p,fl} constructed in §5.1, and both the p-adic norm ||f||^2_{Z_p} and the operator T_{Z_p,N} are defined through this class. The construction requires a class in flat cohomology H^1_fl(X_0(p^n), ((Z/p^n)^\\times)_a), but a class in H^1_fl(Y,G) classifies G-torsors for the fppf topology. The covers π_n: X_1(p^n) → X_0(p^n) are finite flat, but they are not torsors for (Z/p^n)^\\times: the action is free only away from the ramification locus. At cusps (and at supersingular points in characteristic p) the stabilizer is nontrivial for n>1, so X×_Y X is not isomorphic to X×(Z/p^n)^\\times. The text in §5.1 asserts that the fiber product 'is a group scheme over X' and that 'this gives a cochain,' but a ramified cover does not yield a torsor cocycle; the quotient sheaf ((Z/p^n)^\\times)_a is also not defined by the paper. Thus the existence of S_{Z_p,fl} is an unproved assumption, and with it the entire statement of Conjecture 2. The reference to Maz77 Proposition 11.6 only supplies finite flatness in the n=1 case and does not address torsoriality for n>1.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a p-adic analogue of the Harris–Venkatesh derived Hecke algebra for weight-one modular forms. It reviews the mod-p constructions, proves a non-vanishing/uniqueness theorem for a p-adic family of dual forms in completed cohomology under ordinariness (Theorem 4), sketches constructions of p-adic Shimura classes via flat cohomology, and formulates Conjecture 2 asserting that the resulting p-adic norm equals the p-adic regulator of a Stark unit.","tokens_in":11150,"tokens_out":7129,"duration_ms":73980,"significance":"Should Conjecture 2 hold, it would provide a new p-adic complement to the Harris–Venkatesh conjecture, linking derived Hecke actions to p-adic logarithms of Stark units and potentially to p-adic L-functions. The paper also contains a complete, transparent proof of Theorem 4, and it openly distinguishes established results from speculation. However, the central construction of the p-adic Shimura class is not established, and the p-adic regulator is not defined in sufficient detail; these gaps currently prevent Conjecture 2 from being a well-defined mathematical statement.","major_comments":[{"comment":"Section 5.1 claims that the finite flat covers π_n: X_1(p^n) → X_0(p^n) define flat cohomology classes S_{(Z/p^nZ)^×,fl} ∈ H^1_fl(X_0(p^n), ((Z/p^n Z)^×)_a). This is not justified. In fppf cohomology, H^1_fl(Y,G) classifies G-torsors. A group action on a finite flat cover yields a torsor only if the action is free, i.e., X×_Y X ≅ X × G. For n>1, the action of (Z/p^nZ)^× on X_1(p^n) has nontrivial stabilizers at cusps and at supersingular points in characteristic p. The assertion that the fiber product 'is a group scheme over X' and 'gives a cochain' does not produce a torsor cocycle. Hence the existence of S_{Z_p,fl} is unproved, and since ||f||²_{Z_p} (Section 5.3) and T_{Z_p,N} (Section 5.4) are defined through this class, Conjecture 2 is not well-posed as written. This point must be fixed—either by a rigorous construction (for example, via a stack-theoretic or syntomic form of the class) or by explicitly stating the existence as an additional hypothesis.","section":"§5.1"},{"comment":"Section 2 defines the p-adic regulators Reg_{Z_p} and Reg_{Z^×_p} in one sentence by 'omitting the reduction modulo w step' and applying the p-adic logarithm. No proof is given that the result is independent of the choice of place w over p, independent of the branch of the p-adic logarithm, or even that the evaluation at x_{Frob_w} lands in a domain where the p-adic logarithm is defined. Since Conjecture 2 asserts an equality with Reg_{Z_p}(u), this regulator must be a canonically defined map. The manuscript should provide a precise definition and prove its well-definedness, or state the independence as an assumption.","section":"§2"},{"comment":"The sentence in Section 5.3, 'Concretely, what we have done in Section 5 is the construction of the class S_{Z/p^nZ,fl} ∈ H^1(X_{0,1}(p^{n+1},N), Z/p^nZ) using the étale cover X_1(p^n) → X_0(p^n),' contradicts Section 5.1, which states that these covers are not étale for n>1. This internal inconsistency underlines the unresolved status of the flat cohomology construction and must be resolved.","section":"§5.3"}],"minor_comments":[{"comment":"The diagram for the mod-p Shimura class is hard to follow because the steps (a)–(d) are listed after the diagram containing arrows labeled '(a)', etc.; please reorder or add explicit references to the maps. Also, the notation 'Tr Np p' in §3.2 appears garbled; the trace should be written with clear indices.","section":"§3.1"},{"comment":"The quotient sheaf ((Z/p^nZ)^×)_a is introduced without definition; please define its embedding into G_a and the topology used.","section":"§5.1"},{"comment":"The isomorphism ∆ × Z_p ≅ Z_p^× and the decomposition S_{Z_p^×,fl} = S_{∆,fl} × S_{Z_p,fl} are stated without proof; at least indicate how this splitting is compatible with the flat cohomology classes.","section":"§5.2"},{"comment":"Conjecture 2 and the surrounding text use the notation ⟨f^*, T_{Z_p,N}(f)⟩ without specifying the pairing used; the pairing identity in §5.4 defines it for ordinary forms, but the conjecture should restate this.","section":"Conjecture 2"},{"comment":"The paper would benefit from a precise statement of the relation between u and the Stark unit u_f; Remark 3 acknowledges this as a hope, but the unclarity weakens the conjecture's testability.","section":"Remark 3"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is essentially an extended announcement of upcoming work; most of the technical content is deferred. As a self-contained research paper it is not there yet. However, the problem is significant and Theorem 4 is a concrete contribution. I recommend major revision rather than rejection so that the author can either supply the missing foundations or explicitly recast the conjecture as conditional on the existence of the p-adic Shimura class."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a research announcement for a p-adic analogue of the Harris–Venkatesh derived Hecke algebra. The genuinely new content is the proposed p-adic Shimura class and the conjecture (Conjecture 2) that the resulting p-adic norm equals the p-adic Stark regulator. Theorem 4 is a real and clean proof about the projective system of old forms; that part is solid.\n\nThe soft spot is serious. In §5.1, the p-adic Shimura class is constructed by treating the covers π_n: X_1(p^n) → X_0(p^n) as defining flat cohomology classes. But these covers are not torsors for (Z/p^nZ)^× when n>1: the action has nontrivial stabilizers at cusps (and at supersingular points in characteristic p), so X×_Y X is not X×(Z/p^nZ)^×. The paper's claim that 'X×_Y X is a group scheme over X' and 'this gives a cochain' does not supply the needed torsor cocycle. The reference to Mazur's Proposition 11.6 only handles n=1. Without a valid class S_{Z_p,fl}, both the p-adic norm ||f||²_{Z_p} and the operator T_{Z_p,N} are undefined, so Conjecture 2 is not yet a meaningful statement. This is a load-bearing gap, not a cosmetic one.\n\nI don't think the flaw is fatal to the program. The author is transparent that this is upcoming work, and the idea of a Z_p complement to the mod-p Shimura class is worth pursuing. But a referee should insist on a rigorous construction of the flat cohomology class, or a precise formulation using a different topology (or stack-theoretic methods) that accounts for the ramification. The regulator definition in §2 is also terse, though that is a smaller issue.\n\nThe citation pattern looks fine; the paper builds on the cited predecessors and clearly flags dependencies on the author's earlier conjectures. No numerical evidence is claimed, which is appropriate for an announcement.\n\nWho is this for? People working on derived Hecke algebras, Stark units, and p-adic L-functions. I'd want to see the full upcoming paper before citing it, but it deserves a serious referee to sort out the flat cohomology issue.","headline":"A promising p-adic complement to Harris–Venkatesh, with a solid theorem on old forms but a load-bearing gap in the flat cohomology construction that leaves Conjecture 2 undefined.","tokens_in":11739,"tokens_out":4159,"would_cite":false,"duration_ms":36059,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F11","11F33","11F85","11G18","11R42"],"pacs":[],"model":"deepseek-v4-flash","headline":"A p-adic conjecture ties weight-one Hecke actions to Stark-unit logarithms","keywords":["p-adic derived Hecke algebra","weight one modular forms","Shimura classes","Stark units","completed cohomology","flat cohomology","p-adic regulator","Harris–Venkatesh conjecture"],"falsifier":"Write the cochain on $X_1(p^n) \\times_{X_0(p^n)} X_1(p^n)$ for $n > 1$ explicitly and verify it is a flat $1$-cocycle; if it fails, the construction of $S_{\\mathbb{Z}_p,{\\rm fl}}$ collapses. Independently, compute $||f||^2_{\\mathbb{Z}_p}$ via the displayed limit for a concrete dihedral form and compare it with $\\operatorname{Reg}_{\\mathbb{Z}_p}(u)$ for a range of ordinary primes; any mismatch disproves Conjecture 2, while a single matching prime would be genuine evidence.","tokens_in":10450,"feed_emoji":"🧮","tokens_out":9726,"duration_ms":98332,"temperature":0.7,"pith_summary":"This paper tries to move the Harris–Venkatesh derived Hecke algebra story from modulo-$p$ coefficients to the $p$-adic side for weight-one modular forms. It outlines $p$-adic Shimura classes on completed cohomology of modular curves and a $p$-adic derived Hecke operator, then proposes Conjecture 2: for each newform $f$ of weight one, the resulting $p$-adic norm $||f||^2_{\\mathbb{Z}_p}$ equals the $p$-adic regulator of a unit in the Stark unit group attached to the adjoint representation of $f$. If true, the $\\mathbb{Z}_p$-component of the Shimura class completes the mod-$p$ Harris–Venkatesh conjecture and links derived Hecke actions to $p$-adic logarithms of Stark units. Much of the construction, especially the flat-cohomology passage in Section 5.1, is sketched rather than fully proved, so the conjecture is proposed conditionally on that step.","feed_headline":"New p-adic conjecture links Hecke operators to Stark-unit logarithms","feed_subtitle":"A p-adic norm on weight-one forms is predicted to equal a Stark-unit regulator, complementing Harris–Venkatesh.","key_machinery":"The engine is the $p$-adic Shimura class $S_{\\mathbb{Z}_p,{\\rm fl}}$, obtained as the $\\mathbb{Z}_p$-factor of a tower of finite flat covers $X_1(p^n) \\to X_0(p^n)$. Although these covers are only \\'etale at $n=1$, the paper argues they still define classes in flat cohomology $H^1_{\\rm fl}$, which after pushforward and Serre duality become elements of $\\operatorname{Hom}_{\\mathbb{Z}_p}(\\widehat{H}^0(X_0(p^\\infty), \\Omega), \\mathbb{Z}_p)$. The other load-bearing object is the $p$-adic system $\\widehat{f^*}$ of the dual weight-one form: a one-dimensional projective limit in the ordinary case, generated by $\\widehat{f^*}_n = \\alpha^{1-n} f^*_n - \\alpha^{-n} \\omega(p) f^*_{n-1}$, where $\\alpha$ is the unique unit root of $X^2 - a_p X + \\omega(p)p$. Pairing the Shimura class with $f \\cdot \\widehat{f^*}$ defines the $p$-adic norm, and the same construction with a Hecke lifting produces the operator $T_{\\mathbb{Z}_p,N}$ on the ordinary subspace of coherent cohomology.","core_discovery":"On the paper's own terms, the central proposed discovery is a conjectural equality: there exists an element $u \\in U(\\operatorname{Ad}(\\rho_f)) \\otimes \\mathbb{Q}$ and a prime $p_0$ such that for all primes $p \\geq p_0$, $$||f||^2_{\\mathbb{Z}_p} = \\operatorname{Reg}_{\\mathbb{Z}_p}(u),$$ equivalently $\\langle f^*, T_{\\mathbb{Z}_p,N}(f) \\rangle_{\\rm SD} = \\operatorname{Reg}_{\\mathbb{Z}_p}(u)$. The norm is defined as a $p$-adic period pairing the $p$-adic Shimura class $S_{\\mathbb{Z}_p}$ against $f \\cdot \\widehat{f^*}$, where $\\widehat{f^*}$ is built from the $p$-adic system of the dual form by the limit $\\widehat{f^*}_n = \\alpha^{1-n} f^*_n - \\alpha^{-n} \\omega(p) f^*_{n-1}$. This $\\mathbb{Z}_p$-component is complementary to the $(\\mathbb{Z}/p\\mathbb{Z})^\\times$ component of the Harris–Venkatesh conjecture, and together they give a single equality over $\\mathbb{Z}_p^\\times$.","pith_inferences":["Beyond the paper, the conjecture can be tested numerically for dihedral forms: compute $\\operatorname{Reg}_{\\mathbb{Z}_p}(u)$ from the splitting of $p$ in the quadratic field and compare it with the explicit limit formula for $||f||^2_{\\mathbb{Z}_p}$; existing computations for the mod-$p$ case make this a finite check.","Should the flat-cohomology step fail for $n > 1$, a fallback exists: the limit formula in Section 5.3 is written explicitly through $\\mathbb{Z}/p^n\\mathbb{Z}$-cohomology, so the norm could be defined by that limit directly without the non-\\'etale flat classes; the paper does not discuss this possibility.","The author's hope of relating the conjecture to $p$-adic $L$-functions suggests a stronger identity: for dihedral $f$, $\\operatorname{Reg}_{\\mathbb{Z}_p}(u)$ should equal the derivative of a $p$-adic Artin $L$-function at $s=0$, giving the derived Hecke operator a $p$-adic $L$-value interpretation that the paper leaves open."],"forward_implications":["For every ordinary prime $v$ over $p$, Theorem 4 gives a one-dimensional projective limit $\\widehat{f^*}$ in the completed cohomology, so the $p$-adic norm is a specific element of the completed coefficient ring $\\mathcal{O}_v$ rather than an abstract class.","If Conjecture 2 holds, the ordinary subspace carries a $p$-adic derived Hecke operator $T_{\\mathbb{Z}_p,N}$ whose Serre-duality pairing on $(f^*, f)$ is exactly the $p$-adic regulator of a Stark-unit-group element.","Together with the mod-$p$ Harris–Venkatesh conjecture, the $p$-adic statement yields a single equality $||f||^2_{\\mathbb{Z}_p^\\times} = \\operatorname{Reg}_{\\mathbb{Z}_p^\\times}(u)$, combining the torsion and $\\mathbb{Z}_p$ components for all sufficiently large primes.","The construction of $\\widehat{f^*}$ via the unit root $\\alpha$ predicts that only ordinary primes contribute nonzero $p$-adic periods, matching the known arithmetic distinction between ordinary and non-ordinary weight-one forms."],"supporting_citations":[{"why":"Supplies the mod-$p$ derived Hecke operator and the Harris–Venkatesh conjecture that Conjecture 2 complements.","marker":"[HV19]"},{"why":"Defines the Harris–Venkatesh norm, the Stark-unit refinement $u_f$, and the regulator formulation used as the template for Conjecture 2.","marker":"[Zha23b]"},{"why":"Proves the conjecture for dihedral forms and introduces the trace-of-$f(z)f^*(pz)$ construction that the $p$-adic system $\\widehat{f^*}$ generalises.","marker":"[DHRV22]"},{"why":"Constructs the two-dimensional Galois representation $\\rho_f$, from which the adjoint representation and the Stark unit group are formed.","marker":"[DS74]"},{"why":"Provides the modular-curve flatness input for $X_1(p) \\to X_0(p)$ underlying the Shimura covering construction.","marker":"[Maz77]"},{"why":"Gives the Weil-pairing description of the Shimura subgroup, one origin of the Shimura class.","marker":"[Mer96]"},{"why":"Sets out the Stark regulator framework whose $p$-adic variant $\\operatorname{Reg}_{\\mathbb{Z}_p}$ appears in Conjecture 2.","marker":"[Tat84]"}],"fun_headline_variants":["p-adic Hecke conjecture links weight-one forms to Stark logs","p-adic derived Hecke operators meet Stark-unit logarithms","Stark-unit logs predicted for weight-one p-adic Hecke action","p-adic Hecke norm equals Stark regulator on weight-one forms","New p-adic equality: Hecke norm = Stark log for weight-one"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on the claim, sketched in Section 5.1 rather than proved, that the non-\\'etale finite flat covers $X_1(p^n) \\to X_0(p^n)$ for $n > 1$ genuinely produce flat cohomology classes that survive passage to Zariski cohomology and Serre duality; without that, neither the $p$-adic norm nor $T_{\\mathbb{Z}_p,N}$ is defined.","fun_headline_variants_meta":{"raw":{"variants":["p-adic Hecke conjecture links weight-one forms to Stark logs","p-adic derived Hecke operators meet Stark-unit logarithms","Stark-unit logs predicted for weight-one p-adic Hecke action","p-adic Hecke norm equals Stark regulator on weight-one forms","New p-adic equality: Hecke norm = Stark log for weight-one"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000856,"raw_usage":{"total_tokens":3705,"prompt_tokens":918,"completion_tokens":2787,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":534,"completion_tokens_details":{"reasoning_tokens":2694}},"tokens_in":534,"tokens_out":2787,"duration_ms":23472,"temperature":1.0,"reasoning_tokens":2694,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T04:56:46.338334+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Write the cochain on $X_1(p^n) \\times_{X_0(p^n)} X_1(p^n)$ for $n > 1$ explicitly and verify it is a flat $1$-cocycle; if it fails, the construction of $S_{\\mathbb{Z}_p,{\\rm fl}}$ collapses. Independently, compute $||f||^2_{\\mathbb{Z}_p}$ via the displayed limit for a concrete dihedral form and compare it with $\\operatorname{Reg}_{\\mathbb{Z}_p}(u)$ for a range of ordinary primes; any mismatch disproves Conjecture 2, while a single matching prime would be genuine evidence.","supporting_citations":[],"review_version":1}